How can I create a Mobius ring?
The easiest way (I think) to create a Mobius ring is to use a bezier curve with an extrude value.

Add a bezier curve

Extrude and move the curve points into a rough circle.

Increase the 'Extrude' value on the curve properties to give the curve thickness:


The trick is to pick one of the middle curve points and press Ctrl+T to start the 'Tilt' tool. This will allow you to rotate the curve point. Type 180 while using this tool to completely flip the curve point.

You should end up with something like this:



There are actually 4 points in this curve but the last 2 are exactly on top of each other (lower left) to complete the circle.
Convert the curve to a mesh, enter edit mode and merge the start and end vertices using the Alt+M 'Merge' menu.
However, this may cause shading issues as the normals wont know which part of the mesh is 'inside' and which is 'outside'.